In this post, we will see the book Measure, Lebesgue Integrals and Hilbert Space by A. N. Kolmogorov and S. V. Fomin.

About the book:

This publication is the second book. of the “Elements of the Theory of Functions and Functional Analysis,” the first book of which (“Metric and Normed Spaces”) appeared in 1954. In this second book the main role is played by measure theory and the Lebesgue integral. These concepts, in particular the concept of measure, are discussed with a sufficient degree of generality; however, for greater clarity we start with the concept of a Lebesgue measure for plane sets. If the reader so desires he can, having read §1, proceed immediately to Chapter II and then to the Lebesgue integral, taking as the measure, with respect to which the integral is being taken, the usual Lebesgue measure on the line or on the plane.

The theory of measure and of the Lebesgue integral as set forth in this book is based on lectures by A.N. Kolmogorov given by him repeatedly in the Mechanics-Mathematics Faculty of the Moscow State University. The final preparation of the text for publication was carried out by S. V. Fomin.

The two books correspond to the program of the course “Analysis III” which was given for the mathematics students by A. N. Kolmogorov. At the end of this volume the reader will find corrections pertaining to the text of the first volume.